ON THE MOTION OF PENDULUMS. 27 When the fluid is unlimited, it will he found that certain arbitrary constants will vanish by the condition that the motion shall not become infinite at an infinite distance in the fluid. When the fluid is confined by an envelope having a radius J, we have the equations of condition (31). 14. We must now, in accordance with the plan proposed in Section I., introduce the condition that the function ^ shall be composed, so far as the time is concerned, of the circular functions sin nt and cos nt, that is, that it shall be of the form P sin nt 4- Q cos nt, where P and Q are functions of r and 9 only. An artifice, however, which has been extensively employed by M. Cauchy will here be found of great use. Instead of introducing the circular functions sinnt and cos nt, we may employ the exponentials e^"1^ and e~^~Int. Since our equations are linear, and since each of these exponential functions, reproduces itself at each differentiation, it follows that if all the terms in any one of our equations be arranged in two groups, containing as a factor e^~lnt in one case, and e~^~lnt in the other, the two groups will be quite independent, and the equations will be satisfied by either group separately. Hence it will be sufficient to introduce one of the exponential functions. We shall thus have only half the number of terms to write down, and half the number of arbitrary constants to determine that would have been necessary had we employed circular functions. When we have arrived at our result, it will be sufficient to put each equation under the form U+^J— 1F=0, and throw away the imaginary part, or else throw away the real part and omit J — 1, since the system of quantities U, and the system of quantities Fmust separately satisfy the equations of the problem. Assuming then we have to determine P as a function of r and 0.